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Next: Accessing arc labelling information Up: Influence diagrams Previous: Superimposing influence diagrams

Arc influences and diagnostics

 

Influence diagrams contain arcs connecting nodes. We label the arcs as follows. We must first determine what kind of arcs, if any, are to be drawn. This is described in §11.4.1, with the different arc styles summarised in Table 11.4.

We will consider the case where the most detailed standard arcs might be drawn, corresponding to arc style number 7, and so we establish a scenario which might produce such details. The autoarc  control can be used to draw particular arc types automatically between the nodes on an influence diagram, but we will concentrate on tailoring arcs to suit. Suppose that we adjust tex2html_wrap_inline37646 , and produce an influence diagram with arcs connecting the information sources nodes D, E, and F with the node B.

We define the information flows from D, E, and F singly to B as tex2html_wrap_inline37648 , tex2html_wrap_inline37650 and tex2html_wrap_inline37652 respectively, representing the worth of each information source in the absence of any other. The overall resolution at the node B is tex2html_wrap_inline37654 , representing the total information arriving at B. If we observe tex2html_wrap_inline37656 then diagnostic quantities for these resolutions are given by the corresponding size ratios. Thus, for the arc between node D and node B, tex2html_wrap_inline37658 is the ratio given by dividing the observed maximal squared change in adjustment, tex2html_wrap_inline37660 , by its expectation, tex2html_wrap_inline37648 .

We measure the information flow ``from B to D'' by the actual loss in resolution at B if node F is withdrawn from the adjustment. In our notation, this is tex2html_wrap_inline37664 . We measure the information flow from B to E and from B to D similarly, by tex2html_wrap_inline37666 and tex2html_wrap_inline37668 respectively. Diagnostic quantities are also available for these measures; the diagnostic quantity corresponding to tex2html_wrap_inline37668 is tex2html_wrap_inline37672 .

An assessment of how observed information sources work together (or against each other) is given by the path correlation which results when one information source is extracted. Hence, we can label the arc from node D to node B by tex2html_wrap_inline37674 (B), and the other nodes similarly.

The arc styles consist of (1) a straight line connecting two nodes; (2) a box containing information which is placed over the arc; and (3) a circle drawn towards the end of the arc, containing information about the path correlation. We describe the path correlation style first. A path correlation lies in tex2html_wrap_inline37676 . We shade the circle according to the absolute magnitude of the correlation, so that correlations of 1 and -1 result in a fully shaded circle. We shade differently, according to direction. For positive correlations, we shade a corresponding proportion of the circle anti-clockwise, starting from 0 degrees. For negative correlations, we shade a corresponding proportion of the circle clockwise, starting from 0 degrees. The size of the circle can be changed using the pcradius  control.

The box of labelling information consists of a bar divided into two. The half bar nearest the information source concerns information flow from the information source to the node being adjusted, i.e. the contribution of the information source taken singly. The half bar nearest the node being adjusted concerns information flow away from the node being adjusted to the information source, i.e. the loss from extracting the information source singly.

Consider the arc drawn from D to B, where there is no data so that only influences are shown: Table 11.1 shows the contents of an arc label in this case. The region tex2html_wrap_inline37678 represents all the uncertainty in the destination node, proportionately 1. The region tex2html_wrap_inline37680 represents the uncertainty remaining after all information sources have been fitted, tex2html_wrap_inline37682 . The region tex2html_wrap_inline37684 thus represents the proprtion of uncertainty removed in the destination node B. The region tex2html_wrap_inline37686 represents the resolution in uncertainty in B due solely to fitting D, i.e. tex2html_wrap_inline37648 . Thus, when tex2html_wrap_inline37686 is large and tex2html_wrap_inline37692 is small, the implication is that the single source of information D is virtually sufficient. When tex2html_wrap_inline37686 is small and tex2html_wrap_inline37692 is large, the implication instead is that the single source of information D is virtually useless. The outer half of the label is similarly configured: tex2html_wrap_inline37698 is identical to tex2html_wrap_inline37680 , and tex2html_wrap_inline37702 is identical to tex2html_wrap_inline37684 . However, the region tex2html_wrap_inline37706 represents the resolution in uncertainty in B due solely to extracting D from the adjustment, i.e. tex2html_wrap_inline37668 . Thus, when tex2html_wrap_inline37706 is large and tex2html_wrap_inline37712 is small, the implication is that much of the resolution in uncertainty at tex2html_wrap_inline34520 is lost if the single source of information D is withdrawn. When tex2html_wrap_inline37686 is small and tex2html_wrap_inline37692 is large, the implication instead is that the single source of information D contributes little extra to the information supplied already by tex2html_wrap_inline37720 .

 

tex2html_wrap_inline37686 tex2html_wrap_inline37692 tex2html_wrap_inline37680 tex2html_wrap_inline37698 tex2html_wrap_inline37712 tex2html_wrap_inline37706
tex2html_wrap_inline37648 tex2html_wrap_inline37736 tex2html_wrap_inline37682 tex2html_wrap_inline37682 tex2html_wrap_inline37742 tex2html_wrap_inline37668
Table 11.1:  Arc label contents (influences)

Now consider the case where we observe tex2html_wrap_inline37656 , so that we can contrast actual to expected behaviour. The label that we draw is the same, except that we make a further partition of the regions tex2html_wrap_inline37686 and tex2html_wrap_inline37706 in Table 11.1. We obtain a label which resembles that shown in Table 11.2. We partition the region tex2html_wrap_inline37686 according to the magnitude of the size ratio tex2html_wrap_inline37658 . If this ratio is bigger than 1, we shade the portion tex2html_wrap_inline37756 , and otherwise we shade the portion tex2html_wrap_inline37758 . The actual proportions of tex2html_wrap_inline37686 shaded depend on whether or not the ratio is larger than 1. If it is, the proportion of tex2html_wrap_inline37686 shaded is tex2html_wrap_inline37764 , where tex2html_wrap_inline37766 by default, but can be changed using the bigshade  control. If the ratio is smaller than 1, the proportion of tex2html_wrap_inline37686 shaded is tex2html_wrap_inline37770 , where tex2html_wrap_inline37772 by default, but can be changed using the smallshade  control. The region tex2html_wrap_inline37706 is partitioned similarly, except that the pertinent size ratio is tex2html_wrap_inline37672 . The proportions shaded are as described for tex2html_wrap_inline37686 , so that tex2html_wrap_inline37780 is shaded if this ratio is larger than one, and tex2html_wrap_inline37782 is shaded otherwise.

 

tex2html_wrap_inline37756 tex2html_wrap_inline37758 tex2html_wrap_inline37692 tex2html_wrap_inline37680 tex2html_wrap_inline37698 tex2html_wrap_inline37712 tex2html_wrap_inline37782 tex2html_wrap_inline37780
tex2html_wrap_inline37648 tex2html_wrap_inline37736 tex2html_wrap_inline37682 tex2html_wrap_inline37682 tex2html_wrap_inline37742 tex2html_wrap_inline37668
Table 11.2:  Arc label contents (influences and diagnostics)

For both types of label, the labels may be modified by removing the unresolved proportions of uncertainty given by the tex2html_wrap_inline37680 and tex2html_wrap_inline37698 regions. This is achieved by adding 8 to the arc style. All other regions are then scaled up appropriately. This is useful when the actual resolution of uncertainty in the destination is very small, but when it is still important to understand information flow.

It is also possible to draw labels which emphasize the diagnostic information carried by the arc. This is achieved by adding 16 to the arc style. The effect is to obtain labels like those shown in Table 11.2, but with the regions tex2html_wrap_inline37692 , tex2html_wrap_inline37680 , tex2html_wrap_inline37698 , and tex2html_wrap_inline37712 , removed and the remaining regions scaled up.

The length and width of arc labels are set by default, but may be changed by using the arclength  and arcwidth  controls.




next up previous contents index
Next: Accessing arc labelling information Up: Influence diagrams Previous: Superimposing influence diagrams

David Wooff
Wed Oct 21 15:14:31 BST 1998